第四章 : 中纬度的经向环流系统 (III) 授课教师 : 张洋. - Ferrel cell, baroclinic eddies and the westerly jet

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1 第四章 : 中纬度的经向环流系统 (III) - Ferrel cell, baroclinic eddies and the westerly jet 授课教师 : 张洋

2 Outline Review! Observations! The Ferrel Cell!! Review: baroclinic instability and baroclinic eddy life cycle! Eddy-mean flow interaction, E-P flux! Transformed Eulerian Mean equations! Eddy-driven jet! Energy cycle 2

3 - baroclinic instability Review! Baroclinic Instability - is an instability that arises in rotating, stratified fluids that are subject to a horizontal temerature gradient.! Energetics: PE KE low density high temerature B density increasing! Mathematics: density decreasing )) A C high density low temerature! Linear Baroclinic Instability! Linear baroclinic system! Eady s model (1949)! Charney s model (1947) 3

4 - linear baroclinic instability Review! Eady s model (1949)! Charney s model (1947) a) The basic zonal flow has uniform vertical shear, U o (Z) = Z, is a constant b) The fluid is uniformly stratified, N 2 is a constant. The most distinguished difference with Eady s model is that beta effect is considered. c) Two rigid lids at the to and bottom, flat horizontal surface, that is! = 0 at Z = 0 and H. d) The motion is on the f -lane, that is =0 4

5 - linear baroclinic instability Review Linear baroclinic system: Eady model Charney model Small amlitude assumtion Normal mode assumtion Obtain the solutions, e.g. instability conditions growth rate most unstable mode Variable = Basic state + Perturbation u(x,t)=u(z)+u 0 (x,t) u 0 (x,t) U(z) Linearized PV equation (q=pv): t + U x q 0 = 2 0 x y 2 q = 2 y 2 + q 0 + x + f 2 o s y + f 2 o s z q y =0 z s 0 N 2 z s N 2 z 0 i(k x!t) (x,t)=ae Find the conditions for non-trivial solutions and Ci >0 5

6 - linear baroclinic instability! Conclusions: Necessary condition for baroclinic instability: PV gradient changes sign in the interior or boundaries (Charney-stern theory), according to which the midlatitude atmoshere is baroclinic unstable. Different models. i.e. Eady and Charney models have more rigorous conditions. Growth rate: Most unstable mode: = kc i 0.3 f o N k 1 max / L d = Eady in both Eady and Charney models! NH kmax 1 / f o N f o Charney 6

7 - linear baroclinic instability! Eady s model! Charney s model Eric Thomas Eady ( ) 7

8 - linear baroclinic instability JULE CHARNEY was one of the dominant figures in atmosheric science in the three decades following World War II. Much of the change in meteorology from an art to a science is due to his scientific vision and his thorough commitment to eole and rograms in this field. -- by Norman Phillis! Charney s model 8

9 - linear baroclinic instability! Conclusions: Necessary condition for baroclinic instability: PV gradient changes sign in the interior or boundaries (Charney-stern theory), according to which the midlatitude atmoshere is baroclinic unstable. Different models. i.e. Eady and Charney models have more rigorous conditions. Growth rate: Most unstable mode:! Discussion = kc i 0.3 f o N! Normal mode assumtion k 1 max / L d = Eady! Small amlitude assumtion, linearization! Assumtion: uniform vertical shear of the zonal flow in both Eady and Charney models! NH kmax 1 / f o N f o Charney 9

10 - baroclinic eddy life cycle Jet Numerical simulations with idealized GCM: (Thorncroft et al, 1993, Q.J.R.) Basic state at the initial moment: close to real atmoshere. temerature Results: Cature the synotic feature of baroclinic eddies. 10

11 - baroclinic eddy life cycle! Eddies develoment Small amlitude erturbations Wave breaking Finite amlitude erturbations (Thorncroft et al, 1993, Q.J.R.) 11

12 - baroclinic eddy life cycle! Mean flow adjustment Numerical results from Gutowski et al, 1989, JAS, where F and H indicate simulations with friction, diabatic heating, resectively Weaker vertical shear mean reduced temerature gradient. at day 9 Much more stable stratification in the lower trooshere. 12

13 - baroclinic eddy life cycle! Westerly jet and energy cycle: 4 3 C(A Z A E ) Numerical results from Simmons and Hoskins, 1978, JAS Wm -2 u (m s -1 ) Energy conversions C(K Z K E ) C(A E K E ) C(A Z K Z ) Strengthen of westerly jet time (days) 13

14 ! From linear to nonlinear Basic flow or Pre-existing flow (without zonal variation and baroclinic unstable) Small erturbation Linear rocess Reduce the zonal flow temerature gradient; stablize the lower level stratification; enhance the westerly jet Nonlinear interactions Perturbations grow with time (finite amlitude ert.) Eddy-mean interactions (Adjust the zonal flow) Equilibrated states between the adjusted zonal flow and baroclinic eddies 14

15 ! From linear to nonlinear Basic flow or Pre-existing flow (without zonal variation and baroclinic unstable) sinu of eddies MPE ( 10 5 J/m 2 ) Numerical results from a QG model (Zhang, 2009) Nonlinear adjustment Equilibrium EPE ( 10 5 J/m 2 ) Small erturbation day Perturbations grow with time (finite amlitude ert.) Eddy-mean interactions (Adjust the zonal flow) Equilibrated states between the adjusted zonal flow and baroclinic eddies 15

16 ! From linear to nonlinear Basic flow or Pre-existing flow (without zonal variation and baroclinic unstable) sinu of eddies MPE ( 10 5 J/m 2 ) Numerical results from a QG model (Zhang, 2009) Nonlinear adjustment Equilibrium 40 E-P flux 20 EPE ( 10 5 J/m 2 ) Small erturbation day Perturbations grow with time (finite amlitude ert.) Eddy-mean interactions (Adjust the zonal flow) Equilibrated states between the adjusted zonal flow and baroclinic eddies 16

17 Outline! Observations! The Ferrel Cell!! Review: baroclinic instability and baroclinic eddy life cycle! Eddy-mean flow interaction, E-P flux! Transformed Eulerian Mean equations! Eddy-driven jet! The energy cycle 17

18 The Ferrel Cell eddy-zonal flow interaction (I)! Start from the equations:! Momentum equation: du dt fv = x + F x! Continuity equation:! Thermodynamic equation: r v +! =0 d ln dt = Q c T Decomose into zonal mean and eddy comonents: A =[A]+A d = + u + v +! dt t x y 18

19 The Ferrel Cell eddy-zonal flow interaction (I)! The simlified equations:! Momentum equation:! Continuity equation: [u] t = ([u v ]) y [v] y + [!] =0 + f[v]+[f x ]! Thermodynamic equation: [ ] t +[!] s = ([ v ]) y + R/c o [Q] c d = + u + v +! dt t x y Under the quasi-geostrohic aroximation (R o 1) 19

20 - E-P flux! In a steady, adiabatic and frictionless flow:! Momentum equation: [u] t = ([u v ]) y + f[v]+[f x ]! Continuity equation: [v] y + [!] =0! Thermodynamic equation: [ ] t +[!] s = ([ v ]) y + R/c o [Q] c 20

21 - E-P flux! In a QG, steady, adiabatic and frictionless flow:! Momentum equation: f[v] ([u v ]) y =0! Continuity equation: [v] y + [!] =0 r F =0! Thermodynamic equation: [v] = 1 f [!] = y y ([u v ]) [ v ] s / [!] s + ([ v ]) y Define Eliassen-Palm flux: =0 F [u v ] j + f [v ] s / k 21

22 F [u v ] j + f [v ] s / k - E-P flux! In a QG, steady, adiabatic and frictionless flow: [v] = 1 f y ([u v ])! In a QG, steady flow: [!] = y [ v ] s / f[v] r F =0 ([u v ]) +[F x ]=0 y [!] s + ([ v ]) y R/c o [Q] c =0 The meridional overturning flow, in addition to the eddy forcing, has to balance the external forcing. 22

23 F [u v ] j + f [v ] s / k! In a QG, steady flow: Define: [!] =[!]+ y [v] =[v] [v ] s / [v ] s / Residual mean meridional circulation - E-P flux f[v] [!] s + ([ v ]) y [v] y + [!] =0 [ ] 0 t [u] 0 t = [!] s + ([u v ]) +[F x ]=0 y R/c o = f [v]+r F+[F x ] R/c o [Q] c =0 [Q] c 23

24 F [u v ] j + f [v ] s / k! In a QG flow: Define: [!] =[!]+ y [v] =[v] [v ] s / [v ] s / Residual mean meridional circulation - E-P flux [v] y + [!] =0 [ ] 0 t [u] 0 t f[v] [!] s + ([ v ]) y = [!] s + ([u v ]) +[F x ]=0 y R/c o R/c o = f [v]+r F+[F x ] [Q] c (Quasi-geostrohic) Transformed Eulerian mean equations [Q] c =0 24

25 - TEM [ ] t [u] t = [!] R/c s + o = f [v]+r F+[F x ] [v] y + [!] =0 [!] =[!]+ y [v] =[v] [v ] s / [v ] s / [Q] c Recall the streamfunction for the zonal mean flow: m ([v], [!]) =, m y Define a streamfunction for the residual mean circulation: ( [v], [!]) =, y = m + [v ] s /! 25

26 - TEM = m + [v ] s / Case 1: The EADY model Height ( z/d) ψ E (Eulerian) The residual mean circulation s direction can be oosite to the Eulerian mean circulation. Height (z/d) ψ * (Residual) Latitude (y/l) 26

27 Case 2: Observed circulation! In isentroic coordinate The Ferrel Cell D Dt (x, y, z), (x, y, ) D Dt = = t + u r + D Dt Isentroe: An isoleth of entroy. In meteorology it is usually identified with an isoleth of otential temerature. = t + u r + zero for adiabatic flow 27

28 The Ferrel Cell! In isentroic coordinate D Dt = t + u r + D Dt = t + u r + The direction of Ferrel cell is reversed in the isentroic coordinate. (Fig.11.4, Vallis, 2006) 28

29 - TEM = m + [v ] s / Case 2: Observed circulation! In isentroic coordinate D Dt = t + u r + D Dt = t + u r + zero for adiabatic flow The Ferrel cell in the isentroic coordinate is essentially reflect the Residual Mean Circulation. (Fig.11.4, Vallis, 2006) 29

30 ! From linear to nonlinear Basic flow or Pre-existing flow (without zonal variation and baroclinic unstable) Small erturbation Linear rocess Nonlinear interactions E-P flux, residual mean circulation, Transformed Eulerian mean equations. Perturbations grow with time (finite amlitude ert.) Eddy-mean interactions (Adjust the zonal flow) Equilibrated states between the adjusted zonal flow and baroclinic eddies 30

31 - E-P flux: a second view! E-P flux and the Quasi-geostrohic otential vorticity F [u v ] j + f [v ] s / k From the definition of QG otential vorticity: q = 2 y 2 + y + f o 2 1 q 0 = 2 0 x y 2 + f 2 o s 1 0 s PV flux: v 0 q 0 = v v = v( v x = f o 0 v0 s / u y ) y uv x (v2 u 2 ) (u, v) = s = 1 ale 0 0 f o s / y, = alef x o s ale = R R/c o Note: denotes small erturbation v = v v + alef o x v = v + 1 2alef o x 2 thermal wind relation for meridional wind 31

32 - E-P flux: a second view! E-P flux and the Quasi-geostrohic otential vorticity F [u v ] j + f [v ] s / k From the definition of QG otential vorticity: = 1 2 x f o v 0 q 0 = v s / v 02 + y u0 v 0 +f o v 0 0 s / v0 0 u ale 02 s / v = v( v x = v = v u y ) y uv v + alef o x v = v + 1 2alef o x (v2 u 2 ) thermal wind relation for meridional wind x 2 32

33 - E-P flux: a second view! E-P flux and the Quasi-geostrohic otential vorticity F [u v ] j + f [v ] s / k From the definition of QG otential vorticity: v 0 q 0 = v = 1 2 x + y u0 v 0 +f o f o s / v 02 v 0 0 s / v0 0 u ale 02 s / #1 Zonally averaged PV flux by eddies: [v q ] = y [u v ]+f o = r F [v ] s / 33

34 - E-P flux: a second view! E-P flux and the Eliassen-Palm relation F [u v ] j + f [v ] s / k Linearized PV equation (q=pv): t + U q x x 1 2 t [q02 ]+[v 0 q 0 ] q y =0 A = [q02 ] 2 q/y q y =0 Multilying by q and zonally average: Define wave activity density: #2 A t + r F =0 Eliassen-Palm relation 34

35 - E-P flux: a second view! E-P flux and the Rossby waves F [u v ] j + f [v ] s / k Linearized PV equation (q=pv): t + U q x x q y =0 Assume U is fixed, and q y = t + U ale r 2 x 0 + f 2 o Exist solutions of the form 1 0 s 0 i(kx+ly+m!t) =Re e + 0 x =0 Disersion relation of Rossby waves: with grou velocity! = Uk k K 2 K 2 = k 2 + l 2 + m 2 f 2 o /s c gy = 2 kl c K 4 g = 2 kmf o 2 /s K 4 A 0 =ReÂei(kx+ly+m û = Reil, ˆv =Reik!t) ˆ = Reimalef o, ˆq = ReK 2 35

36 - E-P flux: a second view! E-P flux and the Rossby waves F [u v ] j + f [v ] s / k Linearized PV equation (q=pv): t + U q x x with grou velocity q y =0 c gy = 2 kl c K 4 g = 2 kmf o 2 /s K 4 û = K 2 = k 2 + l 2 + m 2 f 2 o /s Reil, ˆv =Reik ˆ = Reimalef o, ˆq = ReK 2 Wave activity density: A = [q02 ] 2 = K4 4 2 [u 0 v 0 ]= 1 2 kl 2 = c gy A f o [v 0 0 ] s / = f 2 o 2s km 2 = c gz A #3 ~F = ~c g A 36

37 E-P flux, TEM and Residual Circulation - Summary! E-P flux: F [u v ] j + f [v ] s / k! In a steady, adiabatic and frictionless flow: #1 #3 [v] = 1 f y ([u v ]) [v q ] = y [u v ]+f o = r F ~F = ~c g A [!] = y! Residual mean circulations: [!] =[!]+ y [v ] s / [ v ] s / [v ] s /, #2 A t + r F =0 A t + r (A ~c g)=0 [v] =[v] r F =0 [v ] s /! TEM equations: [u] t = f [v]+r F+[F x ], [ ] t = [!] s + R/c o [Q] c 37